There are hundreds of tests that need to be performed for certification of a turbine engine.
The FAA requirements and tests are listed in CFR part 33 E and F.
Amongst them are:
- Maximum static thrust tests
- Vibration tests
- Endurance tests
- Water ingestion test
- Hail ingestion test
- Ice cloud test
- Bird strike test
A number of them can be seen in this video of the GE90-115 Engine
The specific requirement for bird ingestion are described in §33.76. The number of birds to be used and their weight depends on the area of the engine inlet.
There have been several changes to the bird ingestion requirements for certification since the A320 engine was certified in 1996. However, once an engine is certified there is no need to demonstrate compliance to the updated requirements; the engine remains certified as it was.
The latest update to these requirements was in 2007, prior to the Hudson accident. To answer your question: no, there have not been any changes to the bird strike testing requirements of engines in the aftermath of the Hudson accident.